A Generalised Spherical Harmonic Deconvolution to Obtain Cubic Texture from Ultrasonic Wave Speed

In this paper, the spherical harmonic convolution approach for HCP materials [1] is extended into a generalised form for the principal purpose of bulk texture determination in cubic polycrystals from ultrasonic wave speed measurements. It is demonstrated that the wave speed function of a general single crystal convolves with the polycrystal Orientation Distribution Function (ODF) to make the resultant polycrystal wave speed function such that when the three functions are expressed in harmonic expansions, the coefficients of any one function may be determined from the coefficients of the other two. All three Euler angles are taken in account in the description of the ODF such that the theorem applies for all general crystal systems. The forward problem of predicting polycrystal wave speed with knowledge of single crystal properties and the ODF is solved for all general cases, with validation carried out on cubic textures showing strong sensitivity to texture and excellent quantitative accuracy in predicted wave speed amplitudes. Importantly, it is also revealed by the theorem that the cubic structure is one of only two crystal systems (the other being HCP) whose orientation distributions can be inversely determined from polycrystal wave velocities by virtue of their respective crystal symmetries. Proof of principle is then established by recovering the ODFs of representative cubic textures solely from the wave velocities generated from a computational model using these texture inputs, and excellent accuracies are achieved in the recovered ODF coefficients as well as the resultant pole figures. Hence the methodology is argued to provide a powerful technique for wave propagation studies and bulk texture measurement in cubic polycrystals and beyond.